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Rational Number Operations

Topic A

Lessons 1 – 4

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Lesson 1: Opposite Quantities Combine to Make Zero

Classwork - Positive and Negative Numbers Review

With your partner, use the graphic organizer below to record what you know about positive and negative numbers. Add

or remove statements during the whole class discussion.

1. Answer the following using the number line as a guide.

a. Where do you begin when locating a number on the number line?

b. What do you call the distance between a number and 0 on a number line?

c. What is the relationship between 7 and −7?

Negative Numbers Positive Numbers

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2. What is the value of the sum of the card values shown? ____________________________________

Use the counting on method on the provided number line to justify your answer.

a. What is the final position on the number line? ________________________________

b. What card or combination of cards would you need to get back to 0? _____________________________

c. How should arrows line up when counting on? ____________________________________________________

3. The Additive Inverse

Use the number line to answer each of the following:

a. How far is 7 from 0 and in which direction? _______________________________

b. What is the opposite of 7? _______________________________

5 -5 -4 8

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c. How far is −7 from 0 and in which direction? _______________________________

d. Thinking back to our previous work, how would you use the counting on method to represent the following: While

playing the Integer Game, the first card selected is 7 and the second card selected is −7?

e. What does this tell us about the sum of 7 and its opposite, −7?

f. Look at the curved arrows you drew for 7 and −7. What relationship exists between these two arrows that would

support your claim about the sum of 7 and −7?

g. Do you think this will hold true for the sum of any number and its opposite?

Exercise 3: Playing the Integer Game

Play the Integer Game with your group. Use a number line to practice counting on.

For all numbers 𝒂 there is a number – 𝒂, such that 𝒂 + (−𝒂) = 𝟎.

The additive inverse of a real number is the opposite of that number on the real number line. For example, the

opposite of −𝟑 is 𝟑. A number and its additive inverse have a sum of 0. The sum of any number and its opposite is

equal to zero.

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Lesson 2: Using the Number Line to Model the Addition of Integers

Classwork

1: Answer the questions below.

a. Suppose you received $10 from your grandmother for your birthday. You spent $4 on snacks. Using addition, how would you write a number sentence to represents this situation?

b. How would you model your equation on a number line to show your answer?

2. Complete the steps to finding the sum of −2 + 3 by filling in the blanks. Model the number sentence using straight arrows called vectors on the number line below.

d. Place the tail of the arrow on _______________.

e. Draw the arrow 2 units to the left of 0, and stop at ________. The direction of the arrow is to the _______ since you are counting down from 0.

f. Start the next arrow at the end of the first arrow, or at _______.

g. Draw the second arrow _______ units to the right since you are counting up from -2.

h. Stop at _______.

i. Repeat the process from part (a) for the expression 3 + (−2).

j. n you say about the sum of −2 + 3 and 3 + (−2)? Does order matter when adding numbers? Why or why not?

g. What can you say about the sum of -2 + 3 and 3 + (-2)? Does order matter when adding numbers? Why or why not?

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3. Modeling absolute value as the length of an arrow.

a. How does absolute value determine the arrow length for −2?

b. How does the absolute value determine the arrow length for 3?

c. How does absolute value help you to represent −10 on a number line?

4. Create a number line model to represent each of the expressions below.

a. −6 + 4

b. 3 + (−8)

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5. Find the sum of the integers represented in the diagram below. Write an equation to express the sum.

a. What three cards are represented in this model? How did you know?

b. In what ways does this model differ from the ones we used in Lesson 1?

c. Can you make a connection between the sum of 6 and where the third arrow ends on the number line?

d. Would the sum change if we changed the order in which we add the numbers, for example, (−2) + 3 + 5?

e. Would the diagram change? If so, how?

−2

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3

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Lesson 3: Understanding Addition of Integers

1: “Counting On” to Express the Sum as Absolute Value on a Number Line

Model of Counting Up Model of Counting Down

Remember that counting up −4 is the same as “the opposite of counting up 4”, and also means counting down 4.

d. For each example above, what is the distance between 2 and the sum?

e. Does the sum lie to the right or left of 2 on a horizontal number line? Vertical number line?

f. Given the expression 54 + 81, can you determine, without finding the sum, the distance between 54 and the sum? Why?

g. Is the sum to the right or left of 54 on the horizontal number line? On a vertical number line?

h. Given the expression 14 + (−3), can you determine, without finding the sum, the distance between 14 and the sum? Why?

i. Is the sum to the right or left of 14 on the number line? On a vertical number line?

2. Work with a partner to create a horizontal number line model to represent each of the

2 + 4 = 6 2 + (−4) = −2

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following expressions. Describe the sum using distance from the 𝑝-value along the number line.

a. −5 + 3

b. −6 + (−2)

c. 7 + (−8)

d. Write an equation, and using the number line, create an “arrow” diagram given the following information:

e. “The 𝑝-value is 6, and the sum lies 15 units to the left of the 𝑝-value.”

Equation:

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Lesson 4: Efficiently Adding Integers and Other Rational Numbers

1. Rule for adding integers with the same signs

a. Represent the sum of 3 + 5 using arrows on the number line.

i. How long is the arrow that represents 𝟑?

ii. What direction does it point?

iii. How long is the arrow that represents 𝟓?

iv. What direction does it point?

v. What is the sum?

vi. If you were to represent the sum using an arrow, how long would the arrow be and what direction would it point?

vii. What is the relationship between the arrow representing the number on the number line and the absolute value of the number?

viii. Do you think that adding two positive numbers will always give you a greater positive number? Why?

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b. Represent the sum of −3 + (−5) using arrows that represent −3 and −5 on the number line. From part (a), use the same questions to elicit feedback. In the Integer Game, I would combine −3 and −5 to give me −8.

i. How long is the arrow that represents −𝟑?

ii. What direction does it point?

iii. How long is the arrow that represents −𝟓?

iv. What direction does it point?

v. What is the sum?

vi. If you were to represent the sum using an arrow, how long would the arrow be and what direction would it point?

vii. Do you think that adding two negative numbers will always give you a smaller negative number? Why?

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c. What do both examples have in common?

RULE: Add integers with the same sign by adding the absolute values and using the common sign.

2. Answer the questions below:

a. Decide whether the sum will be positive or negative without actually calculating the sum.

i. −4 + (−2) ________________________________

ii. 5 + 9 ________________________________

iii. −6 + (−3) ________________________________

iv. −1 + (−11) ________________________________

v. 3 + 5 + 7 ________________________________

vi. −20 + (−15) ________________________________

b. Find the following sums:

i. 15 + 7

ii. −4 + (−16)

iii. −18 + (−64)

iv. −205 + (−123)

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3. Rule for adding integers with opposite signs

a. Represent the 5 + (−3) using arrows on the number line.

How long is the arrow that represents 5?

i. How long is the arrow that represents −𝟑?

ii. What direction does it point?

iii. Which arrow is longer?

iv. What is the sum? If you were to represent the sum using an arrow, how long would the arrow be and what direction would it point?

b. Represent the 4 + (−7) using arrows on the number line.

i. In the two examples above, what is the relationship between length of the arrow representing the sum and the lengths of the arrows representing the 𝒑-value and 𝒒-value?

ii. What is the relationship between the direction of the arrow representing the sum and the direction of arrows representing the 𝒑-value and 𝒒-value?

iii. Write a rule that will give the length and direction of the arrow representing the sum of two values that have opposite signs.

RULE: Add integers with the opposite signs by subtracting the absolute values and using the sign of the integer with the greater absolute value.

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4. Circle the integer with the greater absolute value. Decide whether the sum will be positive or negative without actually calculating the sum.

a. −1 + 2

b. 5 + (−9)

c. −6 + 3

d. −11 + 1

5.Find the following sums:

a. −10 + 7

b. 8 + (−16)

c. −12 + (65)

d. 105 + (−126)

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6. Applying integer addition rules to rational numbers

Find the sum of 6 + (−21

4 ). The addition of rational numbers follows the same rules of addition

for integers.

a. Find the absolute values of the numbers.

b. Subtract the absolute values.

c. The answer will take the sign of the number that has the greater absolute value.

7. Solve the following problems. Show your work.

a. Find the sum of −18 + 7.

b. If the temperature outside was 73 degrees at 5:00 pm, but it fell 19 degrees by 10:00 pm, what is the temperature at 10:00 pm? Write an equation and solve.

c. Write an addition sentence, and find the sum using the diagram below.

−10

31

2

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Lesson Summary

Add integers with the same sign by adding the absolute values and using the common sign.

Steps to adding numbers with opposite signs:

1. Find the absolute values of the numbers.

2. Subtract the absolute values.

3. The answer will take the sign of the number that has the greater absolute value.

To add rational numbers, follow the same rules used to add integers.